4. Principle of calculation#
The displacement \(u(x,t)\) solution of a linear transient dynamic problem can be approximated by its decomposition on a truncated basis of the \({\Phi }_{i}(x)\) eigenmodes:
\(u(x,t)=\sum _{i=1}^{M}{\alpha }_{i}(t){\Phi }_{i}(x)\)
This is what is achieved for example when one treats a transient dynamics problem with the operator DYNA_VIBRA [U4.53.21]. Likewise, we can approach the intensity factors of modal constraints - with the same degree of precision on the result - by the following relationship:
\({K}_{I}(s,t)=\sum _{i=1}^{M}{\alpha }_{i}(t){K}_{I}^{i}(s)\)
where \({\alpha }_{i}(t)\) are the modal contributions, and \({K}_{I}^{i}(s)\) the intensity factors of the modal constraints (function of the curvilinear abscissa \(s\) in 3D, constants in 2D). The intensity factors of the modal constraints are calculated from the natural modes of the structure, using option CALC_K_G of the CALC_G operator.
Since contact is not taken into account, this formula is only valid if the crack remains open at all times. This is generally the case for the applications such as rotating machines (vanes) envisaged, for which centrifugal loading is predominant.
Thus, the operations carried out by the operator POST_K_TRANS are as follows:
recovery in RESU_TRANS of the modal participation factors \({\alpha }_{i}\) from the transitory calculation,
recovery (in TABL_K_MODA) of the intensity factors of the modal constraints,
recombination and printing of dynamic stress intensity factors.
The number \(M\) of modes in the recombination base corresponds, by default, to the number of \({M}^{\mathrm{trans}}\) modes used in the transient calculation. If the number \({M}^{\mathrm{tabl}}\) of modes present in the table TABL_K_MODA provided as input is less than \({M}^{\mathrm{trans}}\), an alarm message is sent and the calculation continues by taking \(M\) equal to \({M}^{\mathrm{tabl}}\).